Combine ½ cup warm water, ¼ cup warm milk (at 108°F), 2¼ teaspoons active dry yeast, and 1 tablespoon granulated sugar in a large measuring cup. Allow this mixture to sit for about 5 minutes until it becomes frothy. If it doesn't foam, start over with fresh yeast.

In a stand mixer bowl, mix together ⅓ cup granulated sugar, ¼ cup melted unsalted butter, 1 egg, 1 egg yolk, and 1 teaspoon vanilla extract. Add 2½ cups all-purpose flour and ½ teaspoon salt. Pour in the activated yeast mixture. Using the dough hook attachment, mix on low until combined, then increase to high speed and knead for 5 minutes. If the dough feels overly sticky, incorporate additional flour, one tablespoon at a time, ensuring it's well mixed after each addition. The dough should remain slightly sticky.

Transfer the dough to a large greased bowl, cover with plastic wrap or a kitchen towel, and let it rise for about an hour or until it doubles in size. Once risen, punch down the dough to release air.

Shape the Donuts:

On a floured surface, roll out the dough to a ½-inch thickness. Use donut or biscuit cutters (4-inch and 1¼-inch diameters recommended) to cut out donuts. Re-roll scraps to maximize the number of donuts.

Place the cut donuts on a parchment-lined baking sheet, cover lightly, and let them rise again for 20-30 minutes.

Preheat the air fryer to 345°F. Lightly coat the air fryer basket with cooking spray. Arrange a few donuts in the basket without touching, spray their tops with cooking spray, and air fry for about 4 minutes. Repeat until all donuts and holes are cooked. Transfer cooked donuts to a paper towel-lined plate.

For the glaze, mix 2 cups powdered sugar, ¼ cup milk, 1 tablespoon light corn syrup, and 1½ teaspoons vanilla extract in a large bowl. While the donuts are still warm, dip them into the glaze and place them on a cooling rack to set. Glazing while warm ensures a shiny finish.